great little image

I always have to double check myself because I can never remember which ones have which

@MichaelT Also the world is apparently roughly shaped like a 5.25" floppy disk

a Magma is a set and binary operation (generally referred to as * or multiplication regardless of it's actual operation) with closure (the binary operation cannot be applied to the set in such a way as to receive a result outside of the set), a Semigroup is a magma where the * is associative which means (a * b) * c = a * (b * c), a monoid is a semigroup with a single identity element i in the set which can multiply with any other element in the set to get back that same element a * i = a

also identity multiplication is commutative so a * i = i * a

inverses are a little trickier, a group is a monoid where every element in the set has a single inverse of it where when multiplied together you get the identity element, and inverses are commutative because an element is the inverse of it's inverse, so for a if it's inverse is b it means that a * b = i and they're inverses of eachother so b * a = i which means they're comutative so a * b = b * a = i